A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of base radius 6 cm. Determine the height of the cone.

Answer

To solve the problem, we need to find the height of the cone formed when a solid metallic hemisphere is melted and recast. We will use the formulas for the volumes of a hemisphere and a cone.

### Step-by-Step Solution:

**Step 1: Calculate the volume of the hemisphere.**

The formula for the volume \( V \) of a hemisphere is given by:

\[ V = \frac{2}{3} \pi r^3 \]

where \( r \) is the radius of the hemisphere. Given that the radius of the hemisphere is 8 cm, we can substitute this value into the formula:

\[ V = \frac{2}{3} \pi (8)^3 \]

Calculating \( (8)^3 \):

\[ (8)^3 = 512 \]

Now substituting back into the volume formula:

\[ V = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi \text{ cm}^3 \]

**Step 2: Set the volume of the cone equal to the volume of the hemisphere.**

The volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone. We know the base radius of the cone is 6 cm, so we can substitute this into the formula:

\[ V = \frac{1}{3} \pi (6)^2 h \]

Calculating \( (6)^2 \):

\[ (6)^2 = 36 \]

Now substituting back into the volume formula for the cone:

\[ V = \frac{1}{3} \pi (36) h = 12 \pi h \text{ cm}^3 \]

**Step 3: Equate the volumes of the hemisphere and the cone.**

Since the hemisphere is melted and recast into the cone, their volumes are equal:

\[ \frac{1024}{3} \pi = 12 \pi h \]

**Step 4: Simplify the equation.**

We can cancel \( \pi \) from both sides:

\[ \frac{1024}{3} = 12h \]

Now, to isolate \( h \), we multiply both sides by \( \frac{1}{12} \):

\[ h = \frac{1024}{3 \times 12} \]

Calculating \( 3 \times 12 \):

\[ 3 \times 12 = 36 \]

So we have:

\[ h = \frac{1024}{36} \]

**Step 5: Simplify \( \frac{1024}{36} \).**

To simplify \( \frac{1024}{36} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 1024 and 36 is 4.

\[ h = \frac{1024 \div 4}{36 \div 4} = \frac{256}{9} \text{ cm} \]

### Final Answer:

The height of the cone is \( \frac{256}{9} \) cm, which is approximately 28.44 cm.